Exponential to Weibull

Can you please help?

Problem. An exponential distribution is a member of the Gamma Family; that is, when , the pdf reduces to the exponential distribution (with ). Let the r.v. Exponential , then , has a Weibull distribution. Show the density function of is

my incorrect solution

This is what I did earlier in the year. Incorporating some of your notation, I think it corresponds to what you showed me above. However, I didn't get the right answer. Can you tell me what I did wrong?